Math 2.0

Group theory and generalizations Finite groups Theorem

Jordan–Hölder theorem

Builds on Composition seriesComposition factors

Statement

Let GG be a group and let {e}=G0⊴⋯⊴Gn=G\{e\} = G_0 \trianglelefteq \cdots \trianglelefteq G_n = G and {e}=H0⊴⋯⊴Hm=G\{e\} = H_0 \trianglelefteq \cdots \trianglelefteq H_m = G be two composition series of GG. Then n=mn = m, and there is a permutation σ\sigma of {1,…,n}\{1, \dots, n\} with Gi/Gi−1≅Hσ(i)/Hσ(i)−1G_i / G_{i-1} \cong H_{\sigma(i)} / H_{\sigma(i) - 1} for each ii.

Remarks

So the composition factors and the composition length of GG are well defined. For G=ℤ/nℤG = \mathbb{Z}/n\mathbb{Z}, the theorem gives the uniqueness of the prime factorisation of nn. One proof uses the Schreier refinement theorem, which says that two subnormal series have equivalent refinements.

History

The theorem is named after Camille Jordan and Otto Hölder.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptConjugate elements, definition, no evidence levelConjugate elementsdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionConjugate subgroup, definition, no evidence levelConjugate subgroupdefinitionLeft coset, definition, no evidence levelLeft cosetdefinitionRight coset, definition, no evidence levelRight cosetdefinitionNormal subgroup, definition, no evidence levelNormal subgroupdefinitionSimple group, definition, no evidence levelSimple groupdefinitionNormality via left and right cosets, statement, no evidence levelNormality via left an…statementQuotient group, definition, no evidence levelQuotient groupdefinitionSubnormal series, definition, no evidence levelSubnormal seriesdefinitionComposition series, definition, no evidence levelComposition seriesdefinitionComposition factors, definition, no evidence levelComposition factorsdefinitionJordan–Hölder theorem, statement, no evidence levelJordan–Hölder theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Group concept
  2. Conjugate elements definition
  3. Subgroup definition
  4. Conjugate subgroup definition
  5. Left coset definition
  6. Right coset definition
  7. Normal subgroup definition
  8. Simple group definition
  9. Normality via left and right cosets statement
  10. Quotient group definition
  11. Subnormal series definition
  12. Composition series definition
  13. Composition factors definition
  14. Jordan–Hölder theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Composition_series, Uniqueness: Jordan–Hölder theorem
About this page
Tag0E088ZP533
URIm2://0E088ZP533
Aliasesjordan-holder-theorem
Kindstatement, theorem
WritingW1 stated
MSC20E15, 20D30
StatusThis copy of the site has no build data. Run m2 status 0E088ZP533.
UsesComposition series, Composition factors

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